Keller’s cube-tiling conjecture is false in high dimensions

Source abstract

O. H. Keller conjectured in 1930 that in any tiling of R n {\mathbb {R}^n} by unit n -cubes there exist two of them having a complete facet in common. O. Perron proved this conjecture for n ≤ 6 n \leq 6 . We show that for all n ≥ 10 n \geq 10 there exists a tiling of R n {\mathbb {R}^n} by unit n -cubes such that no two n -cubes have a complete facet in common.